3.494 \(\int a^{m x} b^{n x} \, dx\)

Optimal. Leaf size=22 \[ \frac{a^{m x} b^{n x}}{m \log (a)+n \log (b)} \]

[Out]

(a^(m*x)*b^(n*x))/(m*Log[a] + n*Log[b])

_______________________________________________________________________________________

Rubi [A]  time = 0.0435366, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{a^{m x} b^{n x}}{m \log (a)+n \log (b)} \]

Antiderivative was successfully verified.

[In]  Int[a^(m*x)*b^(n*x),x]

[Out]

(a^(m*x)*b^(n*x))/(m*Log[a] + n*Log[b])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.16329, size = 22, normalized size = 1. \[ \frac{e^{x \left (m \log{\left (a \right )} + n \log{\left (b \right )}\right )}}{m \log{\left (a \right )} + n \log{\left (b \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a**(m*x)*b**(n*x),x)

[Out]

exp(x*(m*log(a) + n*log(b)))/(m*log(a) + n*log(b))

_______________________________________________________________________________________

Mathematica [A]  time = 0.00717626, size = 22, normalized size = 1. \[ \frac{a^{m x} b^{n x}}{m \log (a)+n \log (b)} \]

Antiderivative was successfully verified.

[In]  Integrate[a^(m*x)*b^(n*x),x]

[Out]

(a^(m*x)*b^(n*x))/(m*Log[a] + n*Log[b])

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 23, normalized size = 1.1 \[{\frac{{a}^{mx}{b}^{nx}}{m\ln \left ( a \right ) +n\ln \left ( b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(a^(m*x)*b^(n*x),x)

[Out]

a^(m*x)*b^(n*x)/(m*ln(a)+n*ln(b))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a^(m*x)*b^(n*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.218419, size = 30, normalized size = 1.36 \[ \frac{a^{m x} b^{n x}}{m \log \left (a\right ) + n \log \left (b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a^(m*x)*b^(n*x),x, algorithm="fricas")

[Out]

a^(m*x)*b^(n*x)/(m*log(a) + n*log(b))

_______________________________________________________________________________________

Sympy [A]  time = 1.02577, size = 42, normalized size = 1.91 \[ \begin{cases} \frac{a^{m x} b^{n x}}{m \log{\left (a \right )} + n \log{\left (b \right )}} & \text{for}\: m \neq - \frac{n \log{\left (b \right )}}{\log{\left (a \right )}} \\b^{n x} x e^{- n x \log{\left (b \right )}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a**(m*x)*b**(n*x),x)

[Out]

Piecewise((a**(m*x)*b**(n*x)/(m*log(a) + n*log(b)), Ne(m, -n*log(b)/log(a))), (b
**(n*x)*x*exp(-n*x*log(b)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223587, size = 439, normalized size = 19.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a^(m*x)*b^(n*x),x, algorithm="giac")

[Out]

2*(2*(m*ln(abs(a)) + n*ln(abs(b)))*cos(-1/2*pi*m*x*sign(a) - 1/2*pi*n*x*sign(b)
+ 1/2*pi*m*x + 1/2*pi*n*x)/((pi*m*sign(a) + pi*n*sign(b) - pi*m - pi*n)^2 + 4*(m
*ln(abs(a)) + n*ln(abs(b)))^2) - (pi*m*sign(a) + pi*n*sign(b) - pi*m - pi*n)*sin
(-1/2*pi*m*x*sign(a) - 1/2*pi*n*x*sign(b) + 1/2*pi*m*x + 1/2*pi*n*x)/((pi*m*sign
(a) + pi*n*sign(b) - pi*m - pi*n)^2 + 4*(m*ln(abs(a)) + n*ln(abs(b)))^2))*e^((m*
ln(abs(a)) + n*ln(abs(b)))*x) - 1/2*I*(-2*I*e^(1/2*I*pi*m*x*sign(a) + 1/2*I*pi*n
*x*sign(b) - 1/2*I*pi*m*x - 1/2*I*pi*n*x)/(I*pi*m*sign(a) + I*pi*n*sign(b) - I*p
i*m - I*pi*n + 2*m*ln(abs(a)) + 2*n*ln(abs(b))) + 2*I*e^(-1/2*I*pi*m*x*sign(a) -
1/2*I*pi*n*x*sign(b) + 1/2*I*pi*m*x + 1/2*I*pi*n*x)/(-I*pi*m*sign(a) - I*pi*n*s
ign(b) + I*pi*m + I*pi*n + 2*m*ln(abs(a)) + 2*n*ln(abs(b))))*e^((m*ln(abs(a)) +
n*ln(abs(b)))*x)